Problem: The complement of an angle is $5^{\circ}$ more than four times the angle. What is the number of degrees in the measure of the angle?
Let the measure of the angle be $x$, so $5^\circ$ more than four times the angle is $4x + 5^\circ$.  Since these two measures are complementary, we have $x + (4x+5^\circ) = 90^\circ$.  Simplifying the left side gives $5x+5^\circ = 90^\circ$, so $5x = 85^\circ$ and $x = \boxed{17^\circ}$.